I am trying to estimate the following model: $$y=B_0 + B_1x_1 + B_2x_2 + B_3x_3 + e$$
However, I have an omitted variable bias because $x_2$ and $x_3$ are not observed.
If I have an (exogenous) instrument $z_1$, that is only expected to affect $x_1$:
$$y=B_0 + B_1az_1 + B_2x_2 + B_3x_3 + e$$
Will I then still get an unbiased estimate of $x_1$?
What happens if $z_1$ might be correlated with one of the omitted variables (let's say $x_3$), but this effect is due to an expected correlation between $x_1$ and $x_3$?
As an example, a policy ($z_1$) affects trust in the government ($x_1$), however because trust in the government leads (causal) to less protests ($x_3$), $z_1$ is necessarily correlated with $x_3$.
$$y=B_0 + B_1TrustinGovernment + B_2x_2 + B_3Protests + e$$ $$y=B_0 + B_1aPolicy + B_2x_2 + B_3Protests + e$$
In essence, this perhaps boils down to the question whether there should be no correlation between $z_1$ and the omitted variables, or that there should be no expected causal effect on the omitted variables.
As a last point,
If $z_1$ might AFFECT one of the omitted variables (let's say $x_2$), is there anything left that can be done? Can I for example control for other variables which I expect to be correlated with the omitted variable (proxy like approach)?